finite subgroup test

For a nonempty finite subset $H$ of a group $G$ , closed under the operation, $H$ is automatically a subgroup -- closure alone forces the identity and every inverse to already be present.

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Examples

Checking that $\{0,3,6,9\}$ is closed under addition mod 12 is already enough to know it's a subgroup of $\mathbb{Z}_{12}$ -- no need to separately check the identity and inverses.